Preheat oven to 350°F. Line two baking sheets with parchment.
Beat butter, granulated sugar, and brown sugar until light and creamy, about 2 minutes.
Mix in beaten egg, vanilla, and peppermint extract until smooth.
Whisk flour, cocoa, baking soda, and salt in a separate bowl.
Add dry ingredients to the butter mixture. Stir just until a soft dough forms. If dry, add milk.
Fold in crushed candy canes and chocolate chips. Chill dough 10–15 minutes for easier shaping.
Scoop 18 portions of dough. Flatten one piece in your palm and place a mini peppermint patty in the center.
Wrap dough around the candy, pinch to seal, and set seam-side down on the sheet, spacing 2 inches apart.
Bake 10–12 minutes, until edges are set and tops look slightly crackly. Do not overbake.
Cool on the sheet 5 minutes, then move to a rack. Sprinkle extra crushed candy canes while warm, if desired.